Sorting

ABSTRACT

The bibliography appearing at the end of this article lists 37 sorting algorithms and 100 books and papers on sorting published in the last 20 years. The basic ideas presented here have been abstracted from this body of work, and the best algorithms known are given as examples. As the algorithms are explained, references to related algorithms and mathematical or experimental analyses are given. Suggestions are then made for choosing the algorithm best suited to a given situation.

REFERENCES

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• 49: Thomas N. Hibbard, An empirical study of minimal storage sorting, Communications of the ACM, v.6 n.5, p.206-213, May 1963 [doi>10.1145/366552.366557]
• 50: Thomas N. Hibbard, A Simple Sorting Algorithm, Journal of the ACM (JACM), v.10 n.2, p.142-150, April 1963 [doi>10.1145/321160.321164]
• 51: Paul Hildebrandt , Harold Isbitz, Radix Exchange—An Internal Sorting Method for Digital Computers, Journal of the ACM (JACM), v.6 n.2, p.156-163, April 1959 [doi>10.1145/320964.320972]
• 52: HOARE, C. A.R. "Quicksort." Computer J. 5, 1 (1962), 10-15.
• 53: HOSKEN, J. C. "Evaluation of sorting methods." In Proe. EJCC, Vol. 8, Spartan Books, New York, Nov. 1955, 39-55.
• 54: George U. Hubbard, Some characteristics of sorting computing systems using random access storage devices, Communications of the ACM, v.6 n.5, p.248-255, May 1963 [doi>10.1145/366552.366586]
• 55: HWANG, F. K.; AND S. LIN. "An analysis of Ford and Johnson's sorting algorithm." In Proc. 3rd Annual Princeton Conf. on Info. Science and Systems, Dept. Electrical Engineering, Princeton Univ., 1969, 292-296.
• 56: E. J. Isaac , R. C. Singleton, Sorting by Address Calculation, Journal of the ACM (JACM), v.3 n.3, p.169-174, July 1956 [doi>10.1145/320831.320834]
• 57: JONES, .R.W. "Sorting by merging." Computer J. 2, 2 (July 1959), 95-96.
• 58: KISLITSYN, S. S. "On finding the kth element of an ordered population by means of pair-wise matching." Sibirsky Matem. Zh. 5, 3 (1964), 557-564.
• 59: Donald E. Knuth , Martin A. Goetz, Letters to the editor: three letters on merging, Communications of the ACM, v.6 n.10, p.585, Oct. 1963 [doi>10.1145/367651.367654]
• 60: Donald E. Knuth, Length of strings for a merge sort, Communications of the ACM, v.6 n.11, p.685-688, Nov. 1963 [doi>10.1145/368310.368397]
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• 66: D. H. Lehmer, Sorting Cards with Respect to a Modulus, Journal of the ACM (JACM), v.4 n.1, p.41-46, Jan. 1957 [doi>10.1145/320856.320865]
• 67: LEVY, S. Y.; AiD M. C. PAULL. "An algebra with application to sorting algorithms." In Proc. 3rd Annual Princeton Conf. on Info. Sciences and Systems, Dept. Electrical Engineering, Princeton Univ., 1969, 286-291.
• 68: LIU, D. "Construction of sorting plans." Presented at Internatl. Symposium on the Theory of Machines and Computations, Haifa, Israel, Aug. 1971.
• 69: LYNCH, W.C. "More combinatorial properties of certain trees." Computer J. 7, 4 (Jan. 1965), 299-302.
• 70: M. Donald MacLaren, Internal Sorting by Radix Plus Sifting, Journal of the ACM (JACM), v.13 n.3, p.404-411, July 1966 [doi>10.1145/321341.321349]
• 71: W. David Malcolm, Jr., String distribution for the polyphase sort, Communications of the ACM, v.6 n.5, p.217-220, May 1963 [doi>10.1145/366552.366560]
• 72: Harold H. Manker, Multiphase sorting, Communications of the ACM, v.6 n.5, p.214-217, May 1963 [doi>10.1145/366552.366559]
• 73: W. A. Martin , D. N. Ness, Optimizing binary trees grown with a sorting algorithm, Communications of the ACM, v.15 n.2, p.88-93, Feb. 1972 [doi>10.1145/361254.361259]
• 74: Armando G. Mendoza, A dispersion pass algorithm for the polyphase merge, Communications of the ACM, v.5 n.10, p.502-504, Oct. 1962 [doi>10.1145/368959.368970]
• 75: MOELLER, D. "MR-1401 a generalized sort program for the card-RAMAC 1401." In IBM Systems Engineering Conf., New York, 1961.
• 76: Robert Morris, Scatter storage techniques, Communications of the ACM, v.11 n.1, p.38-44, Jan. 1968 [doi>10.1145/362851.362882]
• 77: MORRIS, R. "Some theorems on sorting." SIAM J. Appl. Math. 17, 1 (Jan. 1969), 1-6.
• 78: H. Nagler, Amphisbaenic Sorting, Journal of the ACM (JACM), v.6 n.4, p.459-468, Oct. 1959 [doi>10.1145/320998.320999]
• 79: H. Nagler, An estimation of the relative efficiency of two internal sorting methods, Communications of the ACM, v.3 n.11, p.618-620, Nov. 1960 [doi>10.1145/367436.367483]
• 80: J. H. Nichols , A. Tiedrich, Business and management data processing II: A multi-variant generalized sort program employing auxiliary drum storage, Proceedings of the 1962 ACM national conference on Digest of technical papers, p.102-103, September 01-01, 1962 [doi>10.1145/800198.806134]
• 81: O'CONNOR, D.G.;ANDR. J. NELSON. "Sorting system with n-line sorting switch." US Patent 3029413, issued April 10, 1962.
• 82: PAPERNOV, A. A.; AND G. V. STASEVICH. "A method of information sorting in computer memories." Problems of Information Transmission 1, 3 (19677, 63-75.
• 83: J. B. Paterson, The COBOL sorting verb, Communications of the ACM, v.6 n.5, p.255-258, May 1963 [doi>10.1145/366552.366588]
• 84: PICARD, C. "Several recent ideas on the sorting problem." Rev. Franc. Trait. Inf. 9, 1 (1966), 41-46. (French)
• 85: RADKE, C. E. "Merge-sort analysis by matrix techniques." IBM Systems J. 5, 4 (19667, 226-247.
• 86: Samuel W. Reynolds, A generalized polyphase merge algorithm, Communications of the ACM, v.4 n.8, p.347-349, Aug. 1961 [doi>10.1145/366678.366689]
• 87: Samuel W. Reynolds, A generalized polyphase merge algorithm, Communications of the ACM, v.4 n.8, p.347-349, Aug. 1961 [doi>10.1145/366678.366689]
• 88: Thomas Schick, Disk file sorting, Communications of the ACM, v.6 n.6, p.330-331, June 1963 [doi>10.1145/366604.366656]
• 89: SEEBLR, R. R. "Associative self-sorting memory." In Proc. EJCC, Vol. 18, 1960, Spartan Books, N.Y. 179-187.
• 90: D. L. Shell, A high-speed sorting procedure, Communications of the ACM, v.2 n.7, p.30-32, July 1959 [doi>10.1145/368370.368387]
• 91: Donald L. Shell, Optimizing the polyphase sort, Communications of the ACM, v.14 n.11, p.713-719, Nov. 1971 [doi>10.1145/362854.362881]
• 92: Sheldon Sobel, Oscillating Sort—A New Sort Merging Technique, Journal of the ACM (JACM), v.9 n.3, p.372-374, July 1962 [doi>10.1145/321127.321133]
• 93: Edward H. Sussenguth, Jr., Use of tree structures for processing files, Communications of the ACM, v.6 n.5, p.272-279, May 1963 [doi>10.1145/366552.366600]
• 94: Michael E. Tarter , Richard A. Kronmal, Non-uniform key distribution and address calculation sorting, Proceedings of the 1966 21st national conference, p.331-337, January 1966 [doi>10.1145/800256.810712]
• 95: M. H. van Emden, Increasing the efficiency of quicksort, Communications of the ACM, v.13 n.9, p.563-567, Sept. 1970 [doi>10.1145/362736.362753]
• 96: David J. Waks, Conversion, reconversion and comparison techniques in variable-length sorting, Communications of the ACM, v.6 n.5, p.267-272, May 1963 [doi>10.1145/366552.366598]
• 97: WIERZHOWSKI, J. "Sorting by means of random access store." Algorytmy 2, 4 (1965), 59-68.
• 98: WINDLY, P. F. "The influence of storage access time on merging processes in a computer." Computer J. 2, 2 (July 1959), 49-53.
• 99: WINDLY, P. F. "Trees, forests, and rearranging." Computer J. 3, 2 (July 1960), 84-88.
• 100: WOODRUM, L. J. "Internal sorting with minimal comparing." IBM Systems J. 8, 3 (19697, 189-203.
• A1: Philip S. Abrams, Certification of algorithm 245 [M1]: TREESORT, Communications of the ACM, v.8 n.7, p.445, July 1965 [doi>10.1145/364995.365687]
• A2: BATTY, M. A. "Certification of algorithm 201: Shellsort." Comm. A CM 7, 6 (June 19647, 349.
• A3: Charles R. Blair, Certification of Algorithm 207: Stringsort, Communications of the ACM, v.7 n.10, p.585, Oct. 1964 [doi>10.1145/364888.364919]
• A4: Charles R. Blair, Certification of Algorithm 271: Quickersort, Communications of the ACM, v.9 n.5, p.354, May 1966 [doi>10.1145/355592.365613]
• A5: BOOTHROYD, J. "Algorithm 201: Shellsort." Comm. ACM 8, 6 (Aug. 1963,), 445.
• A6: J. Boothroyd, Algorithm 207: stringsort, Communications of the ACM, v.6 n.10, p.615, Oct. 1963 [doi>10.1145/367651.367662]
• A7: BOOTHROYD, J. "Algorithm 25. Sort a section of the elements of an array by determining the rank of each element." Computer J . 10, 3 (Nov. 1967), 308-309.
• A8: BOOTHROYD, J. "Algorithm 26. Order the subscripts of an array section according to the magnitudes of the elements." Computer J . 10, 3 (Nov. 1967), 309-310.
• A9: BOOTHROYD, J. "Algorithm 27. Rearrange the elements of an array section according to a permutation of the subscripts." Computer J. 10, 3 (Nov. 1967), 310.
• A10: J. P. Chandler , W. C. Harrison, Remark on algorithm 201 [M1]: shellsort, Communications of the ACM, v.13 n.6, p.373-374, June 1970 [doi>10.1145/362384.362509]
• A11: Wallace Feurzeig, Algorithm 23: MATH SORT, Communications of the ACM, v.3 n.11, p.601, Nov. 1960 [doi>10.1145/367436.367450]
• A12: Ivan Flores, Algorithm 76: Sorting procedures, Communications of the ACM, v.5 n.1, p.48-50, Jan. 1962 [doi>10.1145/366243.366737]
• A13: Robert W. Floyd, Algorithm 113: Treesort, Communications of the ACM, v.5 n.8, p.434, Aug. 1962 [doi>10.1145/368637.368654]
• A14: Robert W. Floyd, Algorithm 245: Treesort, Communications of the ACM, v.7 n.12, p.701, Dec. 1964 [doi>10.1145/355588.365103]
• A15: Robin Griffin , K. A. Redish, Remark on algorithm 347 [M1]: an efficient algorithm for sorting with minimal storage, Communications of the ACM, v.13 n.1, p.54, Jan. 1970 [doi>10.1145/361953.361993]
• A16: J. S. Hillmore, Certification of Algorithms 63, 64, 65: Partition, quicksort, find, Communications of the ACM, v.5 n.8, p.439, Aug. 1962 [doi>10.1145/368637.368709]
• A17: C. A. R. Hoare, Algorithm 63: partition, Communications of the ACM, v.4 n.7, p.321, July 1961 [doi>10.1145/366622.366642]
• A18: Otto C. Juelich, Remark on algorithm 175: Shuttle sort, Communications of the ACM, v.7 n.5, p.296, May 1964 [doi>10.1145/364099.364321]
• A19: Otto C. Juelich, Remark on algorithm 175: Shuttle sort, Communications of the ACM, v.7 n.5, p.296, May 1964 [doi>10.1145/364099.364321]
• A20: Arthur F. Kaupe, Jr., Algorithm 143: Treesort 1, Communications of the ACM, v.5 n.12, p.604, Dec. 1962 [doi>10.1145/355580.369097]
• A21: Arthur F. Kaupe, Jr., Algorithm 144: Treesort 2:, Communications of the ACM, v.5 n.12, p.604, Dec. 1962 [doi>10.1145/355580.369099]
• A22: Ralph L. London, Certification of algorithm 245 [M1]:treesort 3:proof of algorithms—a new kind of certification, Communications of the ACM, v.13 n.6, p.371-373, June 1970 [doi>10.1145/362384.362507]
• A23: Richard Peto, Remark on algorithm 347: An efficient algorithm for sorting with minimal storage, Communications of the ACM, v.13 n.10, p.624, Oct. 1970 [doi>10.1145/355598.362785]
• A24: B. Randell, Remark on Algorithm 76: Sorting procedures, Communications of the ACM, v.5 n.6, p.348, June 1962 [doi>10.1145/367766.368193]
• A25: B. Randell , L. J. Russell, Certification of algorithms 63, 64 and 65, partition, quicksort, and find, Communications of the ACM, v.6 n.8, p.446, Aug. 1963 [doi>10.1145/366707.367561]
• A26: Russell W. Ranshaw, Certification of algorithm 23: MATHSORT, Communications of the ACM, v.4 n.5, p.238, May 1961 [doi>10.1145/366532.366568]
• A27: George R. Schubert, Certification of algorithm 175: shuttlesort, Communications of the ACM, v.6 n.10, p.619, Oct. 1963 [doi>10.1145/367651.367686]
• A28: R. S. Scowen, Algorithm 271: quickersort, Communications of the ACM, v.8 n.11, p.669-670, Nov. 1965 [doi>10.1145/365660.365678]
• A29: SCOWEN, R. S. "Notes on algorithms 25, 26." Computer J. 12, 4 (Nov. 1969), 408-409.
• A30: C. J. Shaw , T. N. Trimble, Algorithm 175: shuttle sort, Communications of the ACM, v.6 n.6, p.312-313, June 1963 [doi>10.1145/366604.368655]
• A31: Richard C. Singleton, Algorithm 347: an efficient algorithm for sorting with minimal storage [M1], Communications of the ACM, v.12 n.3, p.185-186, March 1969 [doi>10.1145/362875.362901]
• A32: M. H. van Emden, Algorithms 402: Increasing the efficiency of quicksort, Communications of the ACM, v.13 n.11, p.693-694, Nov 1970 [doi>10.1145/362790.362803]
• A33: WILLIAMS, J. W .J . "Algorithm 232: Heapsort." Comm. ACM 7, 6 (June 1964), 347-348.
• A34: WOODALL, A. D. "Algorithm 43: a listed radix sort." Computer J. 12, 4 (Nov. 1969), 406.
• A35: WOODALL, A.D. "Algorithm 45: an internal sorting procedure using a two-way merge." Computer J. 13, 1 (Feb. 1970), 110-111.
• A36: WOODALL, A. D. "Note on algorithms 25, 26." Computer J. 13, 3 (Aug. 1970), 326.
• A37: WOODALL, A.D. "Algorithm 55: an internal merge sort giving ranks of items." Computer J . 13, 4 (Nov. 1970), 424-425.

INDEX TERMS

Primary Classification:
  E. Data
  E.5 FILES
      Subjects: Sorting/searching